There isn’t a precise definition of a forgetful functor.
-
1.
Forgetting Group Structures. The functor $\mathsf{Grp}\to \mathsf{Sets}$ sending a group $\webleft (G,\mu _{G},\eta _{G}\webright )$ to its underlying set $G$, forgetting the multiplication and unit maps $\mu _{G}$ and $\eta _{G}$ of $G$.
-
2.
Forgetting Topologies. The functor $\mathsf{Top}\to \mathsf{Sets}$ sending a topological space $\webleft (X,\mathcal{T}_{X}\webright )$ to its underlying set $X$, forgetting the topology $\mathcal{T}_{X}$.
-
3.
Forgetting Fibrations. The functor $\mathsf{FibSets}\webleft (K\webright )\to \mathsf{Sets}$ sending a $K$-fibred set $\phi _{X}\colon X\to K$ to the set $X$, forgetting the map $\phi _{X}$ and the base set $K$.
-
1.
Forgetting Commutativity. The inclusion functor $\iota \colon \mathsf{CMon}\hookrightarrow \mathsf{Mon}$ which forgets the property of being commutative.
-
2.
Forgetting Inverses. The inclusion functor $\iota \colon \mathsf{Grp}\hookrightarrow \mathsf{Mon}$ which forgets the property of having inverses.
-
3.
Pronunciation of 忘记:
-
•
-
•
Broad IPA transcription: [wɑŋtɕi].
-
•
Sinological IPA transcription: [wɑŋ⁵¹⁻⁵³t͡ɕi⁵¹].
-
•
-
4.
Pronunciation of 遗忘函子:
-
•
-
•
Broad IPA transcription: [iwɑŋ xänt͡sz̩ɨ].
-
•
Sinological IPA transcription: [i³⁵wɑŋ⁵¹ xän³⁵t͡sz̩²¹⁴⁻²¹⁽⁴⁾].
-
•
11.5.3 Forgetful Functors
Despite there not being a formal or precise definition of a forgetful functor, the term is often very useful in practice, similarly to the word “canonical”. The idea is that a “forgetful functor” is a functor that forgets structure or properties, and is best explained through examples, such as the ones below (see Example 11.5.3.1.3 and Example 11.5.3.1.4).
Examples of forgetful functors that forget structure include:
Examples of forgetful functors that forget properties include:
Throughout this work, we will denote forgetful functors that forget structure by ${\text{忘}}$, e.g. as in
\[ {\text{忘}}\colon \mathsf{Grp}\to \mathsf{Sets}. \]
The symbol ${\text{忘}}$, pronounced wasureru (see Item 1 of Remark 11.5.3.1.6 below), means to forget, and is a kanji found in the following words in Japanese and Chinese:
Here we collect the pronunciation of the words in Notation 11.5.3.1.5 for accuracy and completeness.
